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Modeling the Flow of a Bautista-Manero
Fluid in Porous Media
Taha Sochi∗
2009
∗University College London - Department of Physics & Astronomy - Gower Street - London.Email: t.sochi@ucl.ac.uk.
1
Abstract
In this article, the extensional flow and viscosity and the converging-diverging
geometry were examined as the basis of the peculiar viscoelastic behavior in porous
media. The modified Bautista-Manero model, which successfully describes shear-
thinning, elasticity and thixotropic time-dependency, was used for modeling the
flow of viscoelastic materials which also show thixotropic attributes. An algorithm,
originally proposed by Philippe Tardy, that employs this model to simulate steady-
state time-dependent flow was implemented in a non-Newtonian flow simulation
code using pore-scale modeling and the initial results were analyzed. The findings
are encouraging for further future development.
2
Contents
Abstract 2
Contents 3
List of Figures 4
List of Tables 5
1 Introduction 6
2 Viscoelastic Fluids 7
3 Important Aspects for Viscoelastic Flow in Porous Media 10
3.1 Extensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Converging-Diverging Geometry . . . . . . . . . . . . . . . . . . . . 11
4 Modeling the Flow in Porous Media 13
5 Bautista-Manero Model 17
5.1 Tardy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Initial Results of the Modified Tardy Algorithm . . . . . . . . . . . 22
5.2.1 Convergence-Divergence . . . . . . . . . . . . . . . . . . . . 22
5.2.2 Diverging-Converging . . . . . . . . . . . . . . . . . . . . . . 23
5.2.3 Number of Slices . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2.4 Boger Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2.5 Elastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.6 Shear-Thickening . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.7 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2.8 Kinetic Parameter . . . . . . . . . . . . . . . . . . . . . . . 31
6 Conclusions 34
Acknowledgements 35
Nomenclature 36
References 38
Appendix A: Converging-Diverging Geometry 42
3
List of Figures
1 The Tardy algorithm sand pack results forGo=0.1 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with vary-
ing fm (1.0, 0.8, 0.6 and 0.4) . . . . . . . . . . . . . . . . . . . . . . 24
2 The Tardy algorithm Berea results for Go=0.1 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with vary-
ing fm (1.0, 0.8, 0.6 and 0.4) . . . . . . . . . . . . . . . . . . . . . . 24
3 The Tardy algorithm sand pack results forGo=0.1 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with vary-
ing fm (0.8, 1.0, 1.2, 1.4 and 1.6) . . . . . . . . . . . . . . . . . . . 25
4 The Tardy algorithm Berea results for Go=0.1 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with vary-
ing fm (0.8, 1.0, 1.2, 1.4 and 1.6) . . . . . . . . . . . . . . . . . . . 25
5 The Tardy algorithm sand pack results forGo=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, with varying
number of slices for a typical data point (∆P=100 Pa) . . . . . . . 26
6 The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, with varying
number of slices for a typical data point (∆P=200 Pa) . . . . . . . 26
7 The Tardy algorithm sand pack results forGo=1.0 Pa, µ∞=µo=0.1 Pa.s,
λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.6,
0.8, 1.0, 1.2 and 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8 The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=µo=0.1 Pa.s,
λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.6,
0.8, 1.0, 1.2 and 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9 The Tardy algorithm sand pack results for µ∞=0.001 Pa.s, µo=1.0 Pa.s,
λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varyingGo
(0.1, 1.0, 10 and 100 Pa) . . . . . . . . . . . . . . . . . . . . . . . . 29
10 The Tardy algorithm Berea results for µ∞=0.001 Pa.s, µo=1.0 Pa.s,
λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying
Go (0.1, 1.0, 10 and 100 Pa) . . . . . . . . . . . . . . . . . . . . . . 29
11 The Tardy algorithm sand pack results for µ∞=10.0 Pa.s, µo=0.1 Pa.s,
λ=1.0 s, k=10−4 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varyingGo
(0.1 and 100 Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4
5
12 The Tardy algorithm Berea results for µ∞=10.0 Pa.s, µo=0.1 Pa.s,
λ=1.0 s, k=10−4 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying
Go (0.1 and 100 Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
13 The Tardy algorithm sand pack results forGo=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with vary-
ing λ (0.1, 1.0, 10, and 100 s) . . . . . . . . . . . . . . . . . . . . . . 32
14 The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with vary-
ing λ (0.1, 1.0, 10, and 100 s) . . . . . . . . . . . . . . . . . . . . . . 32
15 The Tardy algorithm sand pack results forGo=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=10 s, fe=1.0, fm=0.5, m=10 slices, with varying k
(10−3, 10−4, 10−5, 10−6 and 10−7 Pa−1) . . . . . . . . . . . . . . . . 33
16 The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,
µo=1.0 Pa.s, λ=10 s, fe=1.0, fm=0.5, m=10 slices, with varying k
(10−3, 10−4, 10−5, 10−6 and 10−7 Pa−1) . . . . . . . . . . . . . . . . 33
17 Examples of corrugated capillaries which can be used to model converging-
diverging geometry in porous media . . . . . . . . . . . . . . . . . . 43
18 Radius variation in the axial direction for a corrugated paraboloid
capillary using a cylindrical coordinate system . . . . . . . . . . . . 43
List of Tables
1 Some values of the wormlike micellar system studied by Anderson
and co-workers, which is a solution of surfactant concentrate (a mix-
ture of EHAC and 2-propanol) in an aqueous solution of potassium
chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1 INTRODUCTION 6
1 Introduction
Non-Newtonian fluids are commonly divided into three broad groups [1, 2]:
1. Time-independent: those fluids for which the strain rate at a given point is
solely dependent upon the instantaneous stress at that point.
2. Viscoelastic: those fluids that show partial elastic recovery upon the removal
of a deforming stress.
3. Time-dependent: those fluids for which the strain rate is a function of both
the magnitude and the duration of stress and possibly of the time lapse
between consecutive applications of stress.
A large number of models have been proposed in the literature to model all
types of non-Newtonian fluids under various flow conditions. Most these models
are basically empirical in nature and arising from curve-fitting exercises [3]. In
this article, we investigate a non-Newtonian rheological model (namely Bautista-
Manero model) that can be used to describe viscoelastic behavior with thixotropic
attributes in the context of pore-scale modeling of non-Newtonian flow through
porous media.
2 VISCOELASTIC FLUIDS 7
2 Viscoelastic Fluids
Viscoelastic substances exhibit a dual nature of behavior by showing signs of both
viscous fluids and elastic solids. In its most simple form, viscoelasticity can be
modeled by combining Newton’s law for viscous fluids (stress ∝ rate of strain)
with Hook’s law for elastic solids (stress ∝ strain), as given by the original Maxwell
model and extended by the Convected Maxwell models for the nonlinear viscoelastic
fluids. Although this idealization predicts several basic viscoelastic phenomena, it
does so only qualitatively [4].
Polymeric fluids often show strong viscoelastic effects, which can include shear-
thinning, extension thickening, viscoelastic normal stresses, and time-dependent
rheology phenomena. The equations describing the flow of viscoelastic fluids consist
of the basic laws of continuum mechanics and the rheological equation of state,
or constitutive equation, describing a particular fluid and relates the viscoelastic
stress to the deformation history. The quest is to derive a model that is as simple
as possible, involving the minimum number of variables and parameters, and yet
having the capability to predict the viscoelastic behavior in complex flows [5].
No theory is yet available that can adequately describe all of the observed vis-
coelastic phenomena in a variety of flows. However, many differential and integral
viscoelastic constitutive models have been proposed in the literature. What is com-
mon to all these is the presence of at least one characteristic time parameter to
account for the fluid memory, that is the stress at the present time depends upon
the strain or rate-of-strain for all past times, but with an exponentially fading
memory [6, 7, 8, 9, 10].
The behavior of viscoelastic fluids is drastically different from that of Newtonian
and inelastic non-Newtonian fluids. This includes the presence of normal stresses
in shear flows, sensitivity to deformation type, and memory effects such as stress
relaxation and time-dependent viscosity. These features underlie the observed pe-
culiar viscoelastic phenomena such as rod-climbing (Weissenberg effect), die swell
and open-channel siphon [4, 11]. Most viscoelastic fluids exhibit shear-thinning and
an elongational viscosity that is both strain and extensional strain rate dependent,
in contrast to Newtonian fluids where the elongational viscosity is constant [11].
The behavior of viscoelastic fluids at any time is dependent on their recent
deformation history, that is they possess a fading memory of their past. Indeed
a material that has no memory cannot be elastic, since it has no way of remem-
2 VISCOELASTIC FLUIDS 8
bering its original shape. Consequently, an ideal viscoelastic fluid should behave
as an elastic solid in sufficiently rapid deformations and as a Newtonian liquid in
sufficiently slow deformations. The justification is that the larger the strain rate,
the more strain is imposed on the sample within the memory span of the fluid
[4, 11, 12].
Many materials are viscoelastic but at different time scales that may not be
reached. Dependent on the time scale of the flow, viscoelastic materials mainly
show viscous or elastic behavior. The particular response of a sample in a given
experiment depends on the time scale of the experiment in relation to a natural
time of the material. Thus, if the experiment is relatively slow, the sample will
appear to be viscous rather than elastic, whereas, if the experiment is relatively
fast, it will appear to be elastic rather than viscous. At intermediate time scales
mixed viscoelastic response is observed. Therefore the concept of a natural time
of a material is important in characterizing the material as viscous or elastic. The
ratio between the material time scale and the time scale of the flow is indicated by
a non-dimensional number: the Deborah or the Weissenberg number [3, 13].
A common feature of viscoelastic fluids is stress relaxation after a sudden shear-
ing displacement where stress overshoots to a maximum then starts decreasing
exponentially and eventually settles to a steady state value. This phenomenon
also takes place on cessation of steady shear flow where stress decays over a finite
measurable length of time. This reveals that viscoelastic fluids are able to store
and release energy in contrast to inelastic fluids which react instantaneously to the
imposed deformation [4, 10, 12].
A defining characteristic of viscoelastic materials associated with stress relax-
ation is the relaxation time which may be defined as the time required for the
shear stress in a simple shear flow to return to zero under constant strain condi-
tion. Hence for a Hookean elastic solid the relaxation time is infinite, while for a
Newtonian fluid the relaxation of the stress is immediate and the relaxation time
is zero. Relaxation times which are infinite or zero are never realized in reality
as they correspond to the mathematical idealization of Hookean elastic solids and
Newtonian liquids. In practice, stress relaxation after the imposition of constant
strain condition takes place over some finite non-zero time interval [8].
The complexity of viscoelasticity is phenomenal and the subject is notorious
for being extremely difficult and challenging. The constitutive equations for vis-
coelastic fluids are much too complex to be treated in a general manner. Further
2 VISCOELASTIC FLUIDS 9
complications arise from the confusion created by the presence of other phenomena
such as wall effects and polymer-wall interactions, and these appear to be system
specific. Therefore, it is doubtful that a general fluid model capable of predicting
all the flow responses of viscoelastic system with enough mathematical simplicity
or tractability can emerge in the foreseeable future [10, 14, 15]. Understandably,
despite the huge amount of literature composed in the last few decades on this
subject, almost all these studies have investigated very simple cases in which sub-
stantial simplifications have been made using basic viscoelastic models.
In the last few decades, a general consensus has emerged that in the flow of
viscoelastic fluids through porous media elastic effects should arise, though their
precise nature is unknown or controversial. In porous media, viscoelastic effects
can be important in certain cases. When they are, the actual pressure gradient
will exceed the purely viscous gradient beyond a critical flow rate, as observed
by several investigators. The normal stresses of high molecular polymer solutions
can explain in part the high flow resistances encountered during viscoelastic flow
through porous media. It is argued that the very high normal stress differences and
Trouton ratios (defined as extensional to shear viscosity) associated with polymeric
fluids will produce increasing values of apparent viscosity when flow channels in
the porous medium are of rapidly changing cross section.
Important aspects of non-Newtonian flow in general and viscoelastic flow in
particular through porous media are still presenting serious challenge for modeling
and quantification. There are intrinsic difficulties of characterizing non-Newtonian
effects in the flow of polymer solutions and the complexities of the local geometry
of the porous medium which give rise to a complex and pore-space-dependent
flow field in which shear and extension coexist in various proportions that cannot
be quantified. Flows through porous media cannot be classified as pure shear
flows as the converging-diverging passages impose a predominantly extensional flow
fields especially at high flow rates. Moreover, the extension viscosity of many non-
Newtonian fluids increases dramatically with the extension rate. As a consequence,
the relationship between the pressure drop and flow rate very often do not follow
the observed Newtonian and inelastic non-Newtonian trend. Further complication
arises from the fact that for complex fluids the stress depends not only on whether
the flow is a shearing, extensional, or mixed type, but also on the whole history of
the velocity gradient [5, 16, 17, 18, 19, 20].
3 IMPORTANT ASPECTS FOR VISCOELASTIC FLOW IN POROUS MEDIA10
3 Important Aspects for Viscoelastic Flow in Porous
Media
Strong experimental evidence indicates that the flow of viscoelastic fluids through
packed beds can exhibit rapid increases in the pressure drop, or an increase in
the apparent viscosity, above that expected for a comparable purely viscous fluid.
This increase has been attributed to the extensional nature of the flow field in the
pores caused by the successive expansions and contractions that a fluid element
experiences as it traverses the pore space. Even though the flow field at pore level is
not an ideal extensional flow due to the presence of shear and rotation, the increase
in flow resistance is referred to as an extension thickening effect [19, 21, 22, 23].
There are two major interrelated aspects that have strong impact on the flow
through porous media. These are extensional flow and converging-diverging geom-
etry.
3.1 Extensional Flow
One complexity in the flow in general and through porous media in particular
usually arises from the coexistence of shear and extensional components; sometimes
with the added complication of inertia. Pure shear or elongational flow is very much
the exception in practical situations, especially in the flow through porous media.
By far the most common situation is for mixed flow to occur where deformation
rates have components parallel and perpendicular to the principal flow direction.
In such flows, the elongational components may be associated with the converging-
diverging flow paths [3, 24].
A general consensus has emerged recently that the flow through packed beds has
a substantial extensional component and typical polymer solutions exhibit strain
hardening in extension, which is mainly responsible for the reported dramatic in-
creases in pressure drop. Thus in principle the shear viscosity alone is inadequate to
explain the observed excessive pressure gradients. It is therefore essential to know
the relative importance of elastic and viscous effects or equivalently the relationship
between normal and shear stresses for different shear rates [10, 15, 25].
Elongational flow is fundamentally different from shear, the material property
characterizing the flow is not the viscosity, but the elongational viscosity. The
behavior of the extensional viscosity function is very often qualitatively different
3.2 Converging-Diverging Geometry 11
from that of the shear viscosity function. For example, highly elastic polymer
solutions that possess a viscosity which decreases monotonically in shear often
exhibit an extensional viscosity that increases dramatically with strain rate. Thus,
while the shear viscosity is shear-thinning the extensional viscosity is extension
thickening [3, 5, 13].
3.2 Converging-Diverging Geometry
An important aspect that characterizes the flow in porous media and makes it
distinct from bulk is the presence of converging-diverging flow paths. This geo-
metric factor significantly affects the flow and accentuate elastic responses. Several
arguments have been presented in the literature to explain the effect of converging-
diverging geometry on the flow behavior. Despite this diversity, there is a general
consensus that in porous media the converging-diverging nature of the flow paths
brings out both the extensional and shear properties of the fluid. The principal
mode of deformation to which a material element is subjected as the flow converges
into a constriction involves both a shearing of the material element and a stretch-
ing or elongation in the direction of flow, while in the diverging portion the flow
involves both shearing and compression. The actual channel geometry determines
the ratio of shearing to extensional contributions. In many realistic situations in-
volving viscoelastic flows the extensional contribution is the most important of
the two modes. As porous media flow involves elongational flow components, the
coil-stretch phenomenon can also take place. Consequently, a suitable model pore
geometry is one having converging and diverging sections which can reproduce the
elongational nature of the flow, a feature that is not exhibited by straight capillary
tubes [10, 16, 23, 26, 27].
For long time, the straight capillary tube has been the conventional model for
porous media and packed beds. Despite the general success of this model with
the Newtonian and inelastic non-Newtonian flow, its failure with elastic flow was
remarkable. To redress the flaws of this model, the undulating tube and converging-
diverging channel were proposed in order to include the elastic character of the
flow. Various corrugated tube models with different simple geometries have been
used as a first step to model the effect of converging-diverging geometry on the
flow of viscoelastic fluids in porous media (e.g. [16, 22, 28]). Those geometries
include conically shaped sections, sinusoidal corrugation and abrupt expansions
and contractions. Similarly, a bundle of converging-diverging tubes forms a better
3.2 Converging-Diverging Geometry 12
model for a porous medium in viscoelastic flow than the normally used bundle of
straight capillary tubes, as the presence of diameter variations makes it possible to
account for elongational contributions.
Many investigators have attempted to capture the role of the successive converging-
diverging character of packed bed flow by numerically solving the flow equations
in conduits of periodically varying cross sections. Different opinions on the success
of these models can be found in the literature. Examples include [10, 15, 23, 26,
29, 30]. With regards to modeling viscoelastic flow in regular or random networks
of converging-diverging capillaries, very little work has been done.
4 MODELING THE FLOW IN POROUS MEDIA 13
4 Modeling the Flow in Porous Media
The basic equations describing the flow of fluids consist of the basic laws of con-
tinuum mechanics which are the conservation principles of mass, energy and linear
and angular momentum. These governing equations indicate how the mass, energy
and momentum of the fluid change with position and time. The basic equations
have to be supplemented by a suitable rheological equation of state, or constitutive
equation describing a particular fluid, which is a differential or integral mathemat-
ical relationship that relates the extra stress tensor to the rate-of-strain tensor in
general flow condition and closes the set of governing equations. One then solves
the constitutive model together with the conservation laws using a suitable method
to predict velocity and stress fields of the flows [7, 12, 25, 31, 32, 33].
The mathematical description of the flow in porous media is extremely com-
plex task and involves many approximations. So far, no analytical fluid mechanics
solution to the flow through porous media has been found. Furthermore, such a
solution is apparently out of reach for the foreseeable future. Therefore, to investi-
gate the flow through porous media other methodologies have been developed, the
main ones are the macroscopic continuum approach and the pore-scale numerical
approach.
The widely used continuum models represent a simplified macroscopic approach
in which the porous medium is treated as a continuum. All the complexities and
fine details of the microscopic pore structure are absorbed into bulk terms like
permeability that reflect average properties of the medium. Semi-empirical equa-
tions such as Darcy’s law, Blake-Kozeny-Carman or Ergun equation fall into this
category. Several continuum models are based in their derivation on the capillary
bundle concept. The advantage of the continuum method is that it is simple and
easy to implement with no computational cost. The disadvantage is that it does not
account for the detailed physics at the pore level. One consequence of this is that
in most cases it can only deal with steady-state situations with no time-dependent
transient effects.
In the numerical approach, a detailed description of the porous medium at pore-
scale level is adopted and the relevant physics of flow at this level is applied. To find
the solution, numerical methods, such as finite volume and finite difference, usually
in conjunction with computational implementation are used. The advantage of the
numerical method is that it is the most direct approach to describe the physical
4 MODELING THE FLOW IN POROUS MEDIA 14
situation and the closest to full analytical solution. It is also capable, in principle
at least, to deal with time-dependent transient situations. The disadvantage is that
it requires a detailed pore space description. Moreover, it is usually very complex
and hard to implement and has a huge computational cost with serious convergence
difficulties. Due to these complexities, the flow processes and porous media that
are currently within the reach of numerical investigation are the most simple ones.
Pore-scale network modeling is a relatively novel method developed to deal
with flow through porous media. It can be seen as a compromise between these
two extreme approaches as it partly accounts for the physics and void space de-
scription at the pore level with reasonable and generally affordable computational
cost. Network modeling can be used to describe a wide range of properties from
capillary pressure characteristics to interfacial area and mass transfer coefficients.
The void space is described as a network of pores connected by throats. The pores
and throats are assigned some idealized geometry, and rules which determine the
transport properties in these elements are incorporated in the network to compute
effective transport properties on a mesoscopic scale. The appropriate pore-scale
physics combined with a geologically representative description of the pore space
gives models that can successfully predict average behavior [34, 35].
In our investigation to the flow of Bautista-Manero fluids in porous media we
use network modeling. Our model uses three-dimensional networks built from a
topologically-equivalent three-dimensional voxel image of the pore space with the
pore sizes, shapes and connectivity reflecting the real medium. Pores and throats
are modeled as having triangular, square or circular cross-section by assigning a
shape factor which is the ratio of the area to the perimeter squared and obtained
from the pore space image. Most of the network elements are not circular. To ac-
count for the non-circularity when calculating the volumetric flow rate analytically
or numerically for a cylindrical capillary, an equivalent radius Req is defined:
Req =
(8G
π
)1/4
(1)
where the geometric conductance, G, is obtained empirically from numerical simu-
lation. Two networks obtained from Statoil and representing two different porous
media have been used: a sand pack and a Berea sandstone. These networks are
constructed by Øren and coworkers [36, 37] from voxel images generated by sim-
ulating the geological processes by which the porous medium was formed. The
4 MODELING THE FLOW IN POROUS MEDIA 15
physical and statistical properties of the networks are given in [38].
Assuming a laminar, isothermal and incompressible flow at low Reynolds num-
ber, the only equations that need to be considered are the constitutive equation
for the particular fluid and the conservation of volume as an expression for the
conservation of mass. Because initially the pressure drop in each network element
is not known, an iterative method is used. This starts by assigning an effective vis-
cosity to each network element. The effective viscosity is defined as that viscosity
which makes Poiseulle’s equation fit any set of laminar flow conditions for time-
independent fluids [1]. By invoking the conservation of volume for incompressible
fluid, the pressure field across the entire network is solved using a numerical solver
[39]. Knowing the pressure drops, the effective viscosity of each element is updated
using the expression for the flow rate with a pseudo-Poiseuille definition. The
pressure field is then recomputed using the updated viscosities and the iteration
continues until convergence is achieved when a specified error tolerance in total
flow rate between two consecutive iteration cycles is reached. Finally, the total
volumetric flow rate and the apparent viscosity in porous media, defined as the
viscosity calculated from the Darcy’s law, are obtained.
With regards to modeling the flow in porous media of complex fluids which
have time dependency due to thixotropic or elastic nature, there are three major
difficulties :
• The difficulty of tracking the fluid elements in the pores and throats and
identifying their deformation history, as the path followed by these elements
is random and can have several unpredictable outcomes.
• The mixing of fluid elements with various deformation history in the individ-
ual pores and throats. As a result, the viscosity is not a well-defined property
of the fluid in the pores and throats.
• The change of viscosity along the streamline since the deformation history is
constantly changing over the path of each fluid element.
In the current work, we deal only with one case of steady-state viscoelastic flow,
and hence we did not consider these complications in depth. Consequently, the
tracking of fluid elements or flow history in the network and other dynamic as-
pects are not implemented in the non-Newtonian code. However, time-dependent
effects in steady-state conditions are accounted for in the Tardy algorithm which
is implemented in the code and will be presented in detail in § (5.1).
4 MODELING THE FLOW IN POROUS MEDIA 16
In our modeling approach, to solve the pressure field across a network of n
nodes we write n equations in n unknowns which are the pressure values at the
nodes. The essence of these equations is the continuity of flow of incompressible
fluid at each node in the absence of source and sink. We solve this set of equations
subject to the boundary conditions which are the pressures at the inlet and outlet.
This unique solution is ‘consistent’ and ‘stable’ as it is the only mathematically
acceptable solution to the problem, and, assuming the modeling process and the
mathematical technicalities are correct, should mimic the unique physical reality
of the pressure field in the porous medium.
5 BAUTISTA-MANERO MODEL 17
5 Bautista-Manero Model
This is a relatively simple model that combines the Oldroyd-B constitutive equation
for viscoelasticity and the Fredrickson’s kinetic equation for flow-induced structural
changes usually associated with thixotropy. The model requires six parameters that
have physical significance and can be estimated from rheological measurements.
These parameters are the low and high shear rate viscosities, the elastic modulus,
the relaxation time, and two other constants describing the build up and break
down of viscosity.
The Oldroyd-B model is a simplification of the more elaborate and rarely used
Oldroyd 8-constant model which also contains the upper convected, the lower con-
vected, and the corotational Maxwell equations as special cases. Oldroyd-B is the
second simplest nonlinear viscoelastic model and is apparently the most popular
in viscoelastic flow modeling and simulation. It is the nonlinear equivalent of the
linear Jeffreys model, and hence it takes account of frame invariance in the non-
linear regime. Consequently, in the linear viscoelastic regime the Oldroyd-B model
reduces to the linear Jeffreys model. The Oldroyd-B model can be obtained by
replacing the partial time derivatives in the differential form of the Jeffreys model
with the upper convected time derivatives [12]
τ + λ1
5τ = µo
(γ + λ2
5γ
)(2)
where τ is the extra stress tensor, λ1 is the relaxation time, λ2 is the retardation
time, µo is the low-shear viscosity, γ is the rate-of-strain tensor, and5τ is the upper
convected time derivative of the stress tensor:
5τ =
∂τ
∂t+ v · ∇τ − (∇v)T · τ − τ · ∇v (3)
where t is time, v = (vx, vy, vz) is the fluid velocity vector, (·)T is the transpose of
the tensor and ∇v is the fluid velocity gradient tensor defined by
∇v =
∂vx
∂x∂vx
∂y∂vx
∂z∂vy
∂x
∂vy
∂y
∂vy
∂z∂vz
∂x∂vz
∂y∂vz
∂z
(4)
Similarly,5γ is the upper convected time derivative of the rate-of-strain tensor
5.1 Tardy Algorithm 18
given by:5γ =
∂γ
∂t+ v · ∇γ − (∇v)T · γ − γ · ∇v (5)
The kinetic equation of Fredrickson that accounts for the destruction and con-
struction of structure is given by
dµ
dt=µ
λ
(1− µ
µo
)+ kµ
(1− µ
µ∞
)τ : γ (6)
where µ is the non-Newtonian viscosity, t is the time of deformation, λ is the
relaxation time upon the cessation of steady flow, µo and µ∞ are the viscosities
at zero and infinite shear rates respectively, k is a parameter that is related to a
critical stress value below which the material exhibits primary creep, τ is the stress
tensor and γ is the rate of strain tensor. In this model, λ is a structural relaxation
time, whereas k is a kinetic constant for structure break down. The elastic modulus
Go is related to these parameters by Go = µ/λ [40, 41, 42, 43].
Bautista-Manero model was originally proposed for the rheology of worm-like
micellar solutions which usually have an upper Newtonian plateau, and show strong
signs of shear-thinning. The model, which incorporates shear-thinning, elasticity
and thixotropy, can be used to describe the complex rheological behavior of vis-
coelastic systems that also exhibit thixotropy and rheopexy under shear flow. The
model predicts creep behavior, stress relaxation and the presence of thixotropic
loops when the sample is subjected to transient stress cycles. The Bautista-Manero
model has also been found to fit steady shear, oscillatory and transient measure-
ments of viscoelastic solutions [40, 41, 42, 43].
5.1 Tardy Algorithm
This algorithm is proposed by Philippe Tardy to compute the pressure drop-flow
rate relationship for the steady-state flow of a Bautista-Manero fluid in simple
capillary network models. The bulk rheology of the fluid and the dimensions of
the capillaries making up the network are used as inputs to the models [43]. In the
following paragraphs we outline the basic components of this algorithm and the
logic behind it. This will be followed by some mathematical and technical details
related to the implementation of this algorithm in our non-Newtonian code.
5.1 Tardy Algorithm 19
The flow in a single capillary can be described by the following general relation
Q = G′∆P (7)
where Q is the volumetric flow rate, G′ is the flow conductance and ∆P is the
pressure drop. For a particular capillary and a specific fluid, G′ is given by
G′ = G′(µ) = constant Newtonian Fluid
G′ = G′(µ,∆P ) Purely viscous non-Newtonian Fluid
G′ = G′(µ,∆P, t) Fluid with memory (8)
For a network of capillaries, a set of equations representing the capillaries and
satisfying mass conservation should be solved simultaneously to produce a consis-
tent pressure field, as presented in § (4). For Newtonian fluid, a single iteration
is needed to solve the pressure field since the conductance is known in advance as
the viscosity is constant. For purely viscous non-Newtonian fluid, we start with an
initial guess for the viscosity, as it is unknown and pressure-dependent, and solve
the pressure field iteratively updating the viscosity after each iteration cycle until
convergence is reached. For memory fluids, the dependence on time must be taken
into account when solving the pressure field iteratively. Apparently, there is no
general strategy to deal with such situation. However, for the steady-state flow of
memory fluids a sensible approach is to start with an initial guess for the flow rate
and iterate, considering the effect of the local pressure and viscosity variation due
to converging-diverging geometry, until convergence is achieved. This approach is
adopted by Tardy to find the flow of a Bautista-Manero fluid in a simple capil-
lary network model. In general terms, the Tardy algorithm can be summarized as
follows [43]
• For fluids without memory the capillary is unambiguously defined by its ra-
dius and length. For fluids with memory, where going from one section to
another with different radius is important, the capillary should be modeled
with contraction to account for the effect of converging-diverging geometry
on the flow. The reason is that the effects of fluid memory take place on
going through a radius change, as this change induces a change in shear
rate and generation of extensional flow fields with a consequent viscoelastic
and thixotropic effects. Examples of the converging-diverging geometries are
given in Figure (17).
5.1 Tardy Algorithm 20
• Each capillary is discretized in the flow direction and a discretized form of the
flow equations is used assuming a prior knowledge of the stress and viscosity
at the inlet of the network.
• Starting with an initial guess for the flow rate and using iterative technique,
the pressure drop as a function of the flow rate is found for each capillary.
• Finally, the pressure field for the whole network is found iteratively until
convergence is achieved. Once the pressure field is found the flow rate through
each capillary in the network can be computed and the total flow rate through
the network can be determined by summing and averaging the flow through
the inlet and outlet capillaries.
A modified version of the Tardy algorithm was implemented in our non-Newtonian
code. In the following, we outline the main steps of this algorithm and its imple-
mentation.
• From a one-dimensional steady-state Fredrickson equation in which the par-
tial time derivative is written in the form ∂∂t
= V ∂∂x
, the following equation
can be obtained
Vdµ
dx=µ
λ
(µo − µµo
)+ kµ
(µ∞ − µµ∞
)τ γ (9)
In this formulation, the fluid speed V is given by Q/πr2 and the average shear
rate in the tube with radius r is given by Q/πr3.
• By similar argument, another simplified equation can be obtained from the
Oldroyd-B model
τ +V µ
Go
dτ
dx= 2µγ (10)
• The converging-diverging feature of the capillary is implemented in the form
of a parabolic profile as outlined in Appendix A.
• Each capillary in the network is discretized into m slices, each with width
δx = L/m where L is the capillary length.
5.1 Tardy Algorithm 21
• From Equation (10), applying the simplified assumption dτdx
= (τ2−τ1)δx
where
the subscripts 1 and 2 stand for the inlet and outlet of the slice respectively,
we obtain an expression for τ2 in terms of τ1 and µ2
τ2 =2µ2γ + V µ2τ1
Goδx
1 + V µ2
Goδx
(11)
• From Equations (9) and (11), applying dµdx
= (µ2−µ1)δx
, a third order polynomial
in µ2 is obtained. The coefficients of the four terms of this polynomial are
µ32 : − V
λµoGoδx− 2kγ2
µ∞− kγτ1V
µ∞Goδx
µ22 : − 1
λµo− V 2
Go(δx)2+
V
λGoδx+ 2kγ2 +
kγτ1V
Goδx
µ12 : − V
δx+
1
λ+
V 2µ1
Go(δx)2
µ02 :
V µ1
δx(12)
• The algorithm starts by assuming a Newtonian flow in a network of straight
capillaries. Accordingly, the volumetric flow rate Q, and consequently V and
γ, for each capillary are obtained.
• Starting from the inlet of the network where the viscosity and the stress are
assumed to have known values of µo and R∆P/2L for each capillary respec-
tively, the computing of the non-Newtonian flow in a converging-diverging
geometry takes place in each capillary independently by calculating µ2 and
τ2 slice by slice, where the values from the previous slice are used for µ1 and
τ1 of the current slice.
• For the capillaries which are not at the inlet of the network, the initial values
of the viscosity and stress at the inlet of the capillary are found by computing
the Q-weighted average from all the capillaries that feed into the pore which
is at the inlet of the corresponding capillary.
• To find µ2 of a slice, a bisection numerical method is used. To eliminate
possible non-physical roots, the interval for the accepted root is set between
zero and 3µo with µo used in the case of failure. These conditions are logical
as long as the slice is reasonably thin and the flow and fluid are physically
viable. In the case of a convergence failure, error messages are issued to
5.2 Initial Results of the Modified Tardy Algorithm 22
inform the user. No failure has been detected during the many runs of this
algorithm. Moreover, extensive sample inspection of the µ2 values has been
carried out and proved to be sensible and realistic.
• The value found for the µ2 is used in conjunction with Equation (11) to find
τ2 which is needed as an input to the next slice.
• Averaging the value of µ1 and µ2, the viscosity for the slice is found and used
with Poiseuille law to find the pressure drop across the slice.
• The total pressure drop across the whole capillary is computed by summing
up the pressure drops across its individual slices. This total is used with
Poiseuille law to find the effective viscosity for the capillary as a whole.
• Knowing the effective viscosities for all capillaries of the network, the pressure
field is solved iteratively using an algebraic multi-grid solver, and hence the
total volumetric flow rate from the network and the apparent viscosity are
found.
5.2 Initial Results of the Modified Tardy Algorithm
The modified Tardy algorithm was tested and assessed. Various qualitative aspects
were verified and proved to be correct. Some general results and conclusions are
outlined below with sample graphs and data for the sand pack and Berea net-
works using a calculation box with xl=0.5 and xu=0.95. We would like to remark
that despite our effort to use typical values for the parameters and variables, in
some cases we were forced, for demonstration purposes, to use eccentric values to
accentuate the features of interest. In Table (1) we present some values for the
Bautista-Manero model parameters as obtained from the wormlike micellar system
studied by Anderson and co-workers [44] to give an idea of the parameter ranges
in real fluids. The system is a solution of a surfactant concentrate [a mixture of
the cationic surfactant erucyl bis(hydroxyethyl)methylammonium chloride (EHAC)
and 2-propanol in a 3:2 weight ratio] in an aqueous solution of potassium chloride.
5.2.1 Convergence-Divergence
A dilatant effect due to the converging-diverging feature has been detected relative
to a network of straight capillaries. The viscosity increase is a natural viscoelastic
5.2.2 Diverging-Converging 23
Table 1: Some values of the wormlike micellar system studied by Anderson andco-workers [44] which is a solution of surfactant concentrate (a mixture of EHACand 2-propanol) in an aqueous solution of potassium chloride.
Parameter ValueGo (Pa) 1 - 10µo (Pa.s) 115 - 125µ∞ (Pa.s) 0.00125 - 0.00135λ (s) 1 - 30
k (Pa−1) 10−3
- 10−6
response to the radius tightening at the middle. As the corrugation feature of
the tubes is exacerbated by narrowing the radius at the middle, the viscoelastic
dilatant behavior is intensified. The natural explanation is that the increase in
apparent viscosity should be proportionate to the magnitude of tightening. This
feature is presented in Figures (1) and (2) for the sand pack and Berea networks
respectively on a log-log scale.
5.2.2 Diverging-Converging
The investigation of the effect of diverging-converging geometry by expanding the
radius of the capillaries at the middle revealed a thinning effect relative to the
straight and converging-diverging geometries, as seen in Figures (3) and (4) for
the sand pack and Berea networks respectively on a log-log scale. This is due
to viscoelastic response in the opposite sense to that of converging-diverging by
enlarging the radius at the middle.
5.2.3 Number of Slices
As the number of slices of the capillaries increases, the algorithm converges to a
stable and constant solution within acceptable numerical errors. This indicates
that the numerical aspects of the algorithm are functioning correctly because the
effect of discretization errors is expected to diminish by increasing the number
of slices and hence decreasing their width. A sample graph of apparent viscosity
versus number of slices for a typical data point is presented in Figure (5) for the
sand pack and in Figure (6) for Berea.
5.2.3 Number of Slices 24
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s) Decreasing f m
Figure 1: The Tardy algorithm sand pack results for Go=0.1 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (1.0, 0.8,0.6 and 0.4).
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing f m
Figure 2: The Tardy algorithm Berea results for Go=0.1 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (1.0,0.8, 0.6 and 0.4).
5.2.4 Boger Fluid 25
1.0E-01
1.0E+00
1.0E+01
1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing f m
Figure 3: The Tardy algorithm sand pack results for Go=0.1 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.8, 1.0,1.2, 1.4 and 1.6).
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing f m
Figure 4: The Tardy algorithm Berea results for Go=0.1 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.8,1.0, 1.2, 1.4 and 1.6).
5.2.4 Boger Fluid
A Boger fluid behavior was observed when setting µo = µ∞. However, the apparent
viscosity increased as the converging-diverging feature is intensified. This feature is
5.2.4 Boger Fluid 26
4.1160
4.1180
4.1200
4.1220
4.1240
4.1260
4.1280
4.1300
4.1320
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
Number of Slices
App
aren
t Vis
cosi
ty /
(Pa.
s)
Figure 5: The Tardy algorithm sand pack results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, with varying number of slicesfor a typical data point (∆P=100 Pa).
4.9950
4.9975
5.0000
5.0025
5.0050
5.0075
5.0100
5.0125
5.0150
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
Number of Slices
App
aren
t Vis
cosi
ty /
(Pa.
s)
Figure 6: The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, with varying number of slicesfor a typical data point (∆P=200 Pa).
demonstrated in Figure (7) for the sand pack and in Figure (8) for Berea on a log-
log scale. Since Boger fluid is a limiting and obvious case, this behavior indicates
5.2.5 Elastic Modulus 27
that the model, as implemented, is well-behaved. The viscosity increase is a natural
viscoelastic response to the radius tightening at the middle, as discussed earlier.
5.2.5 Elastic Modulus
The effect of the elastic modulus Go was investigated for shear-thinning fluids,
i.e. µo > µ∞, by varying this parameter over several orders of magnitude while
holding the others constant. It was observed that by increasing Go, the high-
shear apparent viscosities were increased while the low-shear viscosities remained
constant and have not been affected. However, the increase at high-shear rates has
almost reached a saturation point where beyond some limit the apparent viscosities
converged to certain values despite a large increase inGo. A sample of the results for
this investigation is presented in Figure (9) for the sand pack and in Figure (10) for
Berea on a log-log scale. The stability at low-shear rates is to be expected because
at low flow rate regimes near the lower Newtonian plateau the non-Newtonian
effects due to elastic modulus are negligible. As the elastic modulus increases, an
increase in apparent viscosity due to elastic effects contributed by elastic modulus
occurs. Eventually, a saturation will be reached when the contribution of this factor
is controlled by other dominant factors and mechanisms from the porous medium
and flow regime.
5.2.6 Shear-Thickening
A slight shear-thickening effect was observed when setting µo < µ∞ while holding
the other parameters constant. The effect of increasing Go on the apparent vis-
cosities at high-shear rates was similar to the effect observed for the shear-thinning
fluids though it was at a smaller scale. However, the low-shear viscosities are not
affected, as in the case of shear-thinning fluids. A convergence for the appar-
ent viscosities at high-shear rates for large values of Go was also observed as for
shear-thinning. A sample of the results obtained in this investigation is presented
in Figures (11) and (12) for the sand pack and Berea networks respectively on a
log-log scale. It should be remarked that as the Bautista-Manero is originally a
shear-thinning model, it should not be expected to fully-predict shear-thickening
phenomenon. However, the observed behavior is a good sign for our model be-
cause as we push the algorithm beyond its limit, the results are still qualitatively
reasonable.
5.2.6 Shear-Thickening 28
1.0E-02
1.0E-01
1.0E+00
1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing f m
Figure 7: The Tardy algorithm sand pack results for Go=1.0 Pa, µ∞=µo=0.1 Pa.s,λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.6, 0.8, 1.0, 1.2 and1.4).
1.0E-02
1.0E-01
1.0E+00
1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing f m
Figure 8: The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=µo=0.1 Pa.s,λ=1.0 s, k=10−5 Pa−1, fe=1.0, m=10 slices, with varying fm (0.6, 0.8, 1.0, 1.2and 1.4).
5.2.7 Relaxation Time 29
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)Increasing G o
Figure 9: The Tardy algorithm sand pack results for µ∞=0.001 Pa.s, µo=1.0 Pa.s,λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying Go (0.1, 1.0, 10and 100 Pa).
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Increasing G o
Figure 10: The Tardy algorithm Berea results for µ∞=0.001 Pa.s, µo=1.0 Pa.s,λ=1.0 s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying Go (0.1, 1.0, 10and 100 Pa).
5.2.7 Relaxation Time
The effect of the structural relaxation time λ was investigated for shear-thinning
fluids by varying this parameter over several orders of magnitude while holding
5.2.7 Relaxation Time 30
1.0E-01
1.0E+00
1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)G o = 100Pa
G o = 0.1Pa
Figure 11: The Tardy algorithm sand pack results for µ∞=10.0 Pa.s, µo=0.1 Pa.s,λ=1.0 s, k=10−4 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying Go (0.1 and100 Pa).
1.0E-01
1.0E+00
1.0E-06 1.0E-05 1.0E-04 1.0E-03
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
G o = 100Pa
G o = 0.1Pa
Figure 12: The Tardy algorithm Berea results for µ∞=10.0 Pa.s, µo=0.1 Pa.s,λ=1.0 s, k=10−4 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying Go (0.1 and100 Pa).
the others constant. It was observed that by increasing the structural relaxation
time, the apparent viscosities were steadily decreased. However, the decrease at
5.2.8 Kinetic Parameter 31
high-shear rates has almost reached a saturation point where beyond some limit
the apparent viscosities converged to certain values despite a large increase in λ.
A sample of the results is given in Figure (13) for the sand pack and Figure (14)
for Berea on a log-log scale. The effects of relaxation time are expected to ease
as the relaxation time increases beyond a limit such that the effect of interaction
with capillary constriction is negligible. Despite the fact that this feature requires
extensive investigation for quantitative confirmation, the observed behavior seems
qualitatively reasonable considering the flow regimes and the size of relaxation
times of the sample results.
5.2.8 Kinetic Parameter
The effect of the kinetic parameter for structure break down k was investigated for
shear-thinning fluids by varying this parameter over several orders of magnitude
while holding the others constant. It was observed that by increasing the kinetic
parameter, the apparent viscosities generally decreased. However, in some cases
the low-shear viscosities has not been substantially affected. A sample of the results
is given in Figures (15) and (16) for the sand pack and Berea networks respectively
on a log-log scale. The decrease in apparent viscosity on increasing the kinetic
parameter is a natural response as the parameter quantifies structure break down
and hence reflects thinning mechanisms. It is a general trend that the low flow
rate regimes near the lower Newtonian plateau are usually less influenced by non-
Newtonian effects. It will therefore be natural that the low-shear viscosities in
some cases experienced minor changes.
5.2.8 Kinetic Parameter 32
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing λ
Figure 13: The Tardy algorithm sand pack results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying λ (0.1, 1.0,10, and 100 s).
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing λ
Figure 14: The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, k=10−5 Pa−1, fe=1.0, fm=0.5, m=10 slices, with varying λ (0.1, 1.0,10, and 100 s).
5.2.8 Kinetic Parameter 33
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing k
Figure 15: The Tardy algorithm sand pack results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=10 s, fe=1.0, fm=0.5, m=10 slices, with varying k (10−3, 10−4,10−5, 10−6 and 10−7 Pa−1).
1.0E-01
1.0E+00
1.0E+01
1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01
Darcy Velocity / (m/s)
App
aren
t Vis
cosi
ty /
(Pa.
s)
Decreasing k
Figure 16: The Tardy algorithm Berea results for Go=1.0 Pa, µ∞=0.001 Pa.s,µo=1.0 Pa.s, λ=1.0 s, fe=1.0, fm=0.5, m=10 slices, with varying k (10−3, 10−4,10−5, 10−6 and 10−7 Pa−1).
6 Conclusions
A non-Newtonian flow simulation code based on pore-scale network modeling has
been extended to account for viscoelastic and thixotropic behavior using a Bautista-
Manero fluid model. The basis of the implementation is a numerical algorithm
originally suggested by Philippe Tardy. A modified version of this algorithm was
used to investigate the effect of converging-diverging geometry on the steady-
state flow. The implementation of this model has been examined and assessed
using a sand pack and a Berea networks. The generic behavior indicates qual-
itatively correct trends. A conclusion was reached that the current implemen-
tation is a proper basis for investigating the steady-state viscoelastic effects and
some thixotropic effects due to converging-diverging geometries in porous media.
This implementation can be used as a suitable foundation for further develop-
ment. The future work can include elaborating the modified Tardy algorithm and
thoroughly investigating its viscoelastic and thixotropic predictions in quantitative
terms. The code (called Non-Newtonian Code 2) with complete documentation
and the networks can be obtained from this URL: http://www3.imperial.ac.uk/
earthscienceandengineerinresearch/perm/porescalemodellinsoftware/non-newtonian%
20code.
34
Acknowledgements
I would like to thank Dr Valerie Anderson and Dr John Crawshaw of Schlumberger
Cambridge Research Center, and Prof. Martin Blunt and Prof. Geoffrey Maitland
of Imperial College London for their help and advice with regards to various aspects
related to the materials in this article.
35
Nomenclature
γ strain rate (s−1)
γ rate-of-strain tensor
λ structural relaxation time in Fredrickson model (s)
λ1 relaxation time (s)
λ2 retardation time (s)
µ viscosity (Pa.s)
µo zero-shear viscosity (Pa.s)
µ∞ infinite-shear viscosity (Pa.s)
τ stress (Pa)
τ stress tensor
fe scale factor for the entry of corrugated tube (—)
fm scale factor for the middle of corrugated tube (—)
G geometric conductance (m4)
G′ flow conductance (m3.Pa−1.s−1)
Go elastic modulus (Pa)
k parameter in Fredrickson model (Pa−1)
L tube length (m)
P pressure (Pa)
∆P pressure drop (Pa)
Q volumetric flow rate (m3.s−1)
r radius (m)
R tube radius (m)
Req equivalent radius (m)
Rmax maximum radius of corrugated capillary
Rmin minimum radius of corrugated capillary
t time (s)
v fluid velocity vector
∇v fluid velocity gradient tensor
36
V fluid speed (m.s−1)
δx small change in x (m)
5· upper convected time derivative
(·)T matrix transpose
xl
network lower boundary in the non-Newtonian code
xu network upper boundary in the non-Newtonian code
Note: units, when relevant, are given in the SI system. Vectors and tensors are
marked with boldface. Some symbols may rely on the context for unambiguous
identification.
37
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41
Appendix A: Converging-Diverging Geom-
etry and Tube Discretization
There are many simplified converging-diverging geometries for corrugated capillar-
ies which can be used to model the flow of viscoelastic fluids in porous media; some
suggestions are presented in Figure (17). In this study we adopted the paraboloid
and hence developed simple formulae to find the radius as a function of the distance
along the tube axis, assuming cylindrical coordinate system, as shown in Figure
(18).
For the paraboloid depicted in Figure (18), we have
r(z) = az2 + bz + c (13)
Since
r(−L/2) = r(L/2) = Rmax & r(0) = Rmin (14)
the paraboloid is uniquely identified by these three points. On substituting and
solving these equations simultaneously, we obtain
a =
(2
L
)2
(Rmax −Rmin) b = 0 & c = Rmin (15)
that is
r(z) =
(2
L
)2
(Rmax −Rmin)z2 +Rmin (16)
In the modified Tardy algorithm for steady-state viscoelastic flow as imple-
mented in our non-Newtonian code, when a capillary is discretized into m slices
the radius r(z) is sampled at m z-points given by
z = −L2
+ kL
m(k = 1, ...,m) (17)
More complex polynomial and sinusoidal converging-diverging profiles can also
be modeled using this approach.
42
Figure 17: Examples of corrugated capillaries which can be used to modelconverging-diverging geometry in porous media.
z
r
R max R max
R min
L/2-L/2 0
Figure 18: Radius variation in the axial direction for a corrugated paraboloidcapillary using a cylindrical coordinate system.
43
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